The solution set of `4x -3y = 47 and 3x + y =32` is

Options

A. `{15,3}`
B. `{4,12}`
C. `{11,-1}`
D. `{12,3}`

Answer

To solve the system of equations \(4x - 3y = 47\) and \(3x + y = 32\), we will use the method of elimination. Here’s a step-by-step solution:

### Step 1: Write down the equations
We have the following two equations:
1. \(4x - 3y = 47\) (Equation 1)
2. \(3x + y = 32\) (Equation 2)

### Step 2: Eliminate \(y\)
To eliminate \(y\), we can manipulate Equation 2. We will multiply Equation 2 by 3 so that the coefficients of \(y\) in both equations will match.

Multiply Equation 2 by 3:
\[ 3(3x + y) = 3(32) \]
This gives us:
\[ 9x + 3y = 96 \quad \text{(Equation 3)} \]

### Step 3: Add Equations
Now we can add Equation 1 and Equation 3:
\[ (4x - 3y) + (9x + 3y) = 47 + 96 \]
The \(y\) terms will cancel out:
\[ 4x + 9x = 143 \]
This simplifies to:
\[ 13x = 143 \]

### Step 4: Solve for \(x\)
Now, divide both sides by 13:
\[ x = \frac{143}{13} = 11 \]

### Step 5: Substitute \(x\) back to find \(y\)
Now that we have \(x\), we can substitute it back into Equation 2 to find \(y\):
\[ 3(11) + y = 32 \]
This simplifies to:
\[ 33 + y = 32 \]
Now, isolate \(y\):
\[ y = 32 - 33 = -1 \]

### Step 6: Write the solution set
The solution set is:
\[ (x, y) = (11, -1) \]

### Final Answer
The solution set of the equations is \((11, -1)\).

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